I'm not sure how this could possibly apply to people on the lower rungs of any organization. How am I supposed to say no to my graduate advisor or my postdoc advisor? Or, as TT faculty, to that senior member of the dept. or my chair?

This is a totally valid question. And reflects a very real and very difficult situation. Some on how to do this, below. Further:

It seems to me that who is asking is very much a part of the equation and should not be separated from the request.

I disagree with this. Separating them does not mean ignoring who is asking. It's first a matter of deciding about what is being asked. There are things that are obvious "no"s: dog walking, babysitting, and, in my book, coffee fetching. Also, requests for above and beyond wrt hours/timing. For the mentors in the group, this can seem hard. You absolutely need these data for the renewal on which your tenure depends. This student/postdoc/tech is someone who can get it for you. If you need someone to work extra hours, or an experiment somewhat off their main line, it becomes a matter of requesting and discussion about the logistics. Treating trainees with respect and soliciting their help, rather than ordering tends to work best.

Back to trainees: There are things that may not seem right to you, but which you are reluctant to say no. For example, the above situation, the PI asks you do some analysis that is not part of your thesis. It is always possible to try to start a dialogue. "What is involved, why is this important, what is the time frame?". There are ways to ask these that are neutral, and not saying yes or no, but getting more information. If you are upset when the request is first presented, say something like "give me a minute/day and let's talk about this" and walk away. Gather yourself and go back to the discussion when you are not upset. But figure out if you want to do it. If you choose to do it.

As for a TT faculty talking to a senior member: this can be absolutely critical. Senior people, unscrupulous or just plain self-centered/obvlious faculty, can use up a junior person faster than a snowflake on summer asphalt. Especially if you have a method/technique/system that is very attractive or cutting edge or expensive or all three. Junior faculty absolutely must learn to say no, and say it in a non-threatening way. Again, decide whether to do the thing or not, and then figure out how to say no.

There are various strategies for this. It is possible to be blunt and to the point "Your work is really fascinating to me, and I think the questions you are asking are great. But, I am so focused on getting my lab going/getting preliminary data for a grant/getting this last paper out before my renewal is due that I am not going to have time to consider side projects for another year or so. Can we revisit this then?" I favor this as being honest.

Another is "I don't have time to do this in the next few months" or "My tech is too busy to get to this in the next few months" (see, you haven't said "no I won't" but are giving reasons why this is difficult). Follow this with "if you have a tech or student, they can come and shadow mine the next time we do this, and they can learn. But, this may not be for a while". Also, "I don't have the money to do this, can you pay for the stuff it will take?". Those are a bit more dissembling, but they can work. If someone in response to this says to you "But we hired you to collaborate", the best response is "I am eager to extend my work into all these new areas. But until I have established my lab, I can't begin to do that".

Anonymous comment, again:

There are some people who are mature enough to hear the word, "no," and not take it personally. ... Saying no to certain people carries consequences, and to pretend there will be no ramifications because you "did the right thing" is dangerous.

Two points here: one, you need to make not personal. Your response needs not to be about the person to whom you say no, but about what your request/work is. Yes, this is hard. But its part of life.

Second, of course there are ramifications. There are ramifications to everything. This is part of the growing process for whatever you want to do in life, be it PI or otherwise. This is part of human interactions. But if you end up saying "yes" when you mean "no" there will be other ramifications and other consequences, which in the end can be much worse than figuring out how to say no.

Which brings us back to the original point of separation. Decide what you want to do about what is being asked. Going to a meeting when your spouse/partner/offspring/parent is in the hospital (with or without newborn) is a decision about what is important in your life. Saying no to more delicate requests (like run this assay for me), is still a matter of choosing if you want to do the thing. Then, decide how to say no.

Well, first of all, thanks for responding to my comment with a whole post! This topic interests me because it is something that I struggle with. I'm not sure that I understand what this means, though:

"Separating them does not mean ignoring who is asking."

Let's take your example of the PI asking me, the student, to do some data analysis that will help him renew a grant -- a grant that has nothing to do with me or my thesis project. If I "separate" this request from him, then I really have no reason to say yes. I mean, what would be a good a reason, assuming I can't see any possible benefit to me?

So it seems to me that my only reason for doing it is because my PI is asking, and he has been a good, supportive PI to me in the past, and this really is not going to eat up months of my time and set me back significantly on my thesis. And yes, I hope that by being generous with him, he'll be inclined to respond in kind. Maybe he'll send me to an extra conference. Or spend some extra time on my reference letter, who knows? Having a good rapport with those in power over you can be extremely valuable, no?

But again, if I separate the request from the person, what's my reason for saying yes? Is it "doing the right thing" to do uncompensated work for someone? Am I being "compensated" to do this just because I work in his lab? What if he were not paying me as his RA? What if the only reason he wasn't paying me was because he had run out of money? I just don't see how it's possible for me to decide to accept this request or not without taking into account the person who is asking and what their behavior has been with me in the past.

Sadly, I can't really trust my gut on this. Because my gut will always tell me I should be nice, helpful, etc. So I will always want to help.... And certain people will always try to take advantage of that.

Anonymous, "assuming I can't see any possible benefit to me" + "Having a good rapport with those in power over you can be extremely valuable."

I think you answered your own question. There is be a benefit to you. It is not direct in the sense that the data analysis you do will not end up in your thesis, but there is a benefit (assuming the requester is trustworthy).

The idea of separating the request from the requester, as I understand it, is to help you to objectively evaluate the amount of effort involved in the request along with the amount of time you have available to complete the request. By (attempting) to remove your personal feelings of the requester, you can more clearly answer those questions. Then your response can incorporate that information. As Potnia said, you may not say "no" but instead let the requester know the parameters of your assistance.

I said no to my PI once. He wanted me to do an additional 8 hours of assays after I had already done 12+ continuous hours of tissue processing. I was exhausted and aching, my arms were trembling from exertion and I was fed up. So I said "I will run those assays tomorrow. If I try to run them tonight I will make mistakes and drop things and ruin the whole experiment. If there is a valid scientific reason why it has to be done tonight, fine, but you need to know that the probability of a serious mistake will go way up."

And he backed down, and let me go home.

But my situation was unusual: I was a tech, not a grad student. I had a union that (I hoped) would back me up. And just the week before, there was national news coverage of a woman grad student who had been murdered while working alone late at night.

The PI never asked me to do anything like that (more than a 12 hour day) again. I don't know if it would have hurt my career, but he lost the grant about a year later so I left.